Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. II.
type: Item
instance of: scholarly article
title: Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. II.
zbMATH Open document ID: 1086.35041
DOI: 10.1016/S0294-1449(03)00013-1
author: Patrizia Pucci
author: James Serrin
author: Filippo Gazzola
author: Lambertus A. Peletier
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2003-11-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78606
review text: In the present study the authors deal with the ground states of the quasilinear elliptic equation \[ -\Delta_mu=-\delta u^{m-1}+u^{p-1} \text{ in }\mathbb{R}^N\tag{1} \] in which \(1<m<N\), \(m<p<m^*\), \(\delta> 0\) and \(m^*\) is the critical Sobolev exponent \(m^*=\frac{Nm}{N-m}\). Here by \(\Delta_m\) is denoted \(\Delta_mu:=\text{div}(|\nabla u|^{m-2}\nabla u)\). The authors mainly interested in a radially symmetric solutions for (1). The authors present the exact asymptotic behaviour of the ground state, for any \(1<m<N\), both at the origin and outside the origin, when the equation tends to critical growth. Moreover, the authors obtain the ``equilibrium'' relation between \(\varepsilon(p=m^*-\varepsilon)\) and \(\delta\) so that when they both vanish according to this relation, ground states neither blow up nor vanish. Part I, cf. ibid. 19, No. 4, 477--504 (2002; Zbl 1013.35031), Part III, J. Differ. Equations 198, No. 1, 53--90 (2004; Zbl 1046.35043).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2009152
zbMATH Keywords: \(m\)-Laplacian
zbMATH Keywords: Ground states
zbMATH Keywords: Shooting methods
zbMATH Keywords: Asymptotic behavior
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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OpenAlex ID: W4205700382
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description: scientific article; zbMATH DE number 2009152